×

zbMATH — the first resource for mathematics

Functional mixed effects wavelet estimation for spectra of replicated time series. (English) Zbl 1395.62285
Summary: Motivated by spectral analysis of replicated brain signal time series, we propose a functional mixed effects approach to model replicate-specific spectral densities as random curves varying about a deterministic population-mean spectrum. In contrast to existing work, we do not assume the replicate-specific spectral curves to be independent, i.e. there may exist explicit correlation between different replicates in the population. By projecting the replicate-specific curves onto an orthonormal wavelet basis, estimation and prediction is carried out under an equivalent linear mixed effects model in the wavelet coefficient domain. To cope with potentially very localized features of the spectral curves, we develop estimators and predictors based on a combination of generalized least squares estimation and nonlinear wavelet thresholding, including asymptotic confidence sets for the population-mean curve. We derive \(L_{2}\)-risk bounds for the nonlinear wavelet estimator of the population-mean curve – a result that reflects the influence of correlation between different curves in the replicate population – and consistency of the estimators of the inter- and intra-curve correlation structure in an appropriate sparseness class of functions. To illustrate the proposed functional mixed effects model and our estimation and prediction procedures, we present several simulated time series data examples and we analyze a motivating brain signal dataset recorded during an associative learning experiment.
MSC:
62M15 Inference from stochastic processes and spectral analysis
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G08 Nonparametric regression and quantile regression
62G15 Nonparametric tolerance and confidence regions
62P10 Applications of statistics to biology and medical sciences; meta analysis
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Abramovich, F., Benjamini, Y., Donoho, D. L. and Johnstone, I. M. (2006). Adapting to unknown sparsity by controlling the false discovery rate., The Annals of Statistics 34 584-653. · Zbl 1092.62005
[2] Antoniadis, A. and Sapatinas, T. (2007). Estimation and inference in functional mixed-effects models., Computational Statistics & Data Analysis 51 4793-4813. · Zbl 1162.62341
[3] Aston, J., Chiou, J. M. and Evans, J. P. (2010). Linguistic pitch analysis using functional principal component mixed effect models., Journal of the Royal Statistical Society: Series C 59 297-317.
[4] Brillinger, D. R. (1981)., Time Series: Data Analysis and Theory . Holden-Day, San Francisco. · Zbl 0486.62095
[5] Bruce, A. G. and Gao, H. Y. (1996). Understanding Waveshrink: Variance and bias estimation., Biometrika 83 727-745. · Zbl 0883.62038
[6] Diggle, P. J. and Al Wasel, I. (1997). Spectral analysis of replicated biomedical time series., Journal of the Royal Statistical Society: Series C 46 31-71. · Zbl 0893.62108
[7] Fiecas, M. and Ombao, H. (2016). Modeling the evolution of dynamic brain processes during an associative learning experiment., Journal of the American Statistical Association . (Accepted). · Zbl 1232.62147
[8] Freyermuth, J. M., Ombao, H. andvon Sachs, R. (2010). Tree-structured wavelet estimation in a mixed effects model for spectra of replicated time series., Journal of the American Statistical Association 105 634-646. · Zbl 1392.62282
[9] Gao, H. Y. (1997). Choice of thresholds for wavelet shrinkage estimate of the spectrum., Journal of Time Series Analysis 18 231-251. · Zbl 0923.62100
[10] Genovese, C. R. and Wasserman, L. (2005). Confidence sets for nonparametric wavelet regression., The Annals of Statistics 698-729. · Zbl 1068.62057
[11] Giacofci, M., Lambert-Lacroix, S., Marot, G. and Picard, F. (2013). Wavelet-based clustering for mixed-effects functional models in high dimension., Biometrics 69 31-40. · Zbl 1274.62774
[12] Gorrostieta, C., Ombao, H., Prado, R., Patel, S. and Eskandar, E. (2012). Exploring dependence between brain signals in a monkey during learning., Journal of Time Series Analysis 33 771-778. · Zbl 1281.92012
[13] Guo, W. (2002). Functional mixed effects models., Biometrics 58 121-128. · Zbl 1209.62072
[14] Hernandez-Flores, C., Artiles-Romero, J. and Saavedra-Santana, P. (1999). Estimation of the population spectrum with replicated time series., Computational Statistics & Data Analysis 30 271-280. · Zbl 1042.62605
[15] Higham, N. J. (2002). Computing the nearest correlation matrix; a problem from finance., IMA Journal of Numerical Analysis 22 329-343. · Zbl 1006.65036
[16] Iannaccone, R. and Coles, S. (2001). Semiparametric models and inference for biomedical time series with extra-variation., Biostatistics 2 261-276. · Zbl 1097.62561
[17] Jiang, J. (2007)., Linear and Generalized Linear Mixed Models and Their Applications . Springer, New York. · Zbl 1152.62040
[18] Jiang, J., Luan, Y., Wang, Y. et al. (2007). Iterative estimating equations: Linear convergence and asymptotic properties., The Annals of Statistics 35 2233-2260. · Zbl 1126.62025
[19] Johnstone, I. M. (2015). Gaussian Estimation: Sequence and Multiresolution Models. (Unpublished, manuscript).
[20] Krafty, R. T. (2016). Discriminant analysis of time series in the presence of within-group spectral variability., Journal of Time Series Analysis 37 435-450. · Zbl 1359.62374
[21] Krafty, R. T., Hall, M. and Guo, W. (2011). Functional mixed effects spectral analysis., Biometrika 98 583-598. · Zbl 1231.62168
[22] Krafty, R. T., Rosen, O., Stoffer, D. S., Buysse, D. J. and Hall, M. H. (2016). Conditional spectral analysis of replicated multiple time series with application to nocturnal physiology., arXiv
[23] Martinez, J. G., Bohn, K. M., Carroll, R. J. and Morris, J. S. (2013). A study of Mexican free-tailed bat chirp syllables: Bayesian functional mixed models for nonstationary acoustic time series., Journal of the American Statistical Association 108 514-526. · Zbl 06195957
[24] Morris, J. S. (2014). Functional regression., arXiv · Zbl 1366.57013
[25] Morris, J. S. and Carroll, R. J. (2006). Wavelet-based functional mixed models., Journal of the Royal Statistical Society: Series B 68 179-199. · Zbl 1110.62053
[26] Morris, J. S., Brown, P. J., Herrick, R. C., Baggerly, K. A. and Coombes, K. R. (2008). Bayesian Analysis of mass spectrometry proteomic data using wavelet-based functional mixed models., Biometrics 64 479-489. · Zbl 1137.62399
[27] Moulin, P. (1994). Wavelet thresholding techniques for power spectrum estimation., IEEE Transactions on Signal Processing 42 3126-3136.
[28] Nason, G. (2010)., Wavelet Methods in Statistics with R . Springer, New York. · Zbl 1165.62033
[29] Neumann, M. H. (1996). Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series., Journal of Time Series Analysis 17 601-633. · Zbl 0873.62099
[30] Neumann, M. H. and von Sachs, R. (1997). Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra., The Annals of Statistics 25 38-76. · Zbl 0871.62081
[31] Ombao, H., von Sachs, R. and Guo, W. (2005). SLEX analysis of multivariate nonstationary time series., Journal of the American Statistical Association 100 519-531. · Zbl 1117.62407
[32] Pav, S. E. (2015). Moments of the log non-central chi-square distribution., arXiv
[33] Qin, L. and Guo, W. (2006). Functional mixed-effects model for periodic data., Biostatistics 7 225-234. · Zbl 1169.62386
[34] Qin, L., Guo, W. and Litt, B. (2009). A time-frequency functional model for locally stationary time series data., Journal of Computational and Graphical Statistics 18 675-693.
[35] Robins, J. andvan der Vaart, A. W. (2006). Adaptive nonparametric confidence sets., The Annals of Statistics 34 229-253. · Zbl 1091.62039
[36] Rudzkis, R., Saulis, L. and Statulevińćius, V. (1978). A general lemma on probabilities of large deviations., Lithuanian Mathematical Journal 18 226-238. · Zbl 0423.60027
[37] Searle, S. R., Casella, G. and McCulloch, C. E. (1992)., Variance Components . John Wiley & Sons, New Jersey. · Zbl 0850.62007
[38] Taniguchi, M. (1979). On estimation of parameters of Gaussian stationary processes., Journal of Applied Probability 16 575-591. · Zbl 0417.60048
[39] van der Vaart, A. W. (2000)., Asymptotic Statistics . Cambridge University Press, Cambridge U.K. · Zbl 0910.62001
[40] Vidakovic, B. (1999)., Statistical Modeling by Wavelets . John Wiley & Sons, New York. · Zbl 0924.62032
[41] Wahba, G. (1980). Automatic smoothing of the log periodogram., Journal of the American Statistical Association 75 122-132. · Zbl 0442.62074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.